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Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns
Accepted Manuscript
Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns Ting-Uei Lee, Xiaochen Yang, Jiayao Ma, Yan Chen, Joseph M. Gattas PII: DOI: Reference:
S0020-7403(18)33185-0 https://doi.org/10.1016/j.ijmecsci.2018.11.005 MS 4631
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
27 September 2018 5 November 2018 6 November 2018
Please cite this article as: Ting-Uei Lee, Xiaochen Yang, Jiayao Ma, Yan Chen, Joseph M. Gattas, Elastic buckling shape control of thin-walled cylinder using pre-embedded curvedcrease origami patterns, International Journal of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.11.005
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Highlights • A new elastic buckling control method for thin-walled cylinder is presented.
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• The failure mode is pre-determined as a stabilised high-order elastica surface.
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• The deformed surface is measured and shown to have a near-exact correspondence.
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Graphical Abstract
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Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns
Key Laboratory of Mechanism and Equipment Design of Ministry of Education, Tianjin, China b School of Mechanical Engineering, Tianjin University, Tianjin, China c School of Civil Engineering, University of Queensland, St. Lucia, Australia
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Ting-Uei Leea,b,c , Xiaochen Yanga,b , Jiayao Maa,b,∗, Yan Chena,b , Joseph M. Gattasb
Abstract
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Decades of research has led to a comprehensive understanding of the buckling behaviour of thin-walled tubes. Many of these studies have attempted to control the buckling-behaviour of thin-walled tubes by utilising their imperfection sensitive characteristics to guide the deformation process to a predictable buckling mode. However, a key limitation of such techniques is an inability to predict the exact deformed shape of post-buckled tubes. This study presents a new method to control the shape of a elastically buckled medium length thin-walled cylinder by using pre-embedded curved-crease origami patterns. The failure mode is pre-determined as a stabilised highorder elastica surface, which manifests via a diamond buckling mode. A set of prototypes are tested and show that the buckling process can be guided to a range of designed failure modes. The deformed surface is measured and shown to have a near-exact correspondence to the analytical description, where the average absolute surface error is less than half of the 0.3mm sheet thickness. This study then closely explores the driving mechanics of the buckling process and shows that the controllable buckling process exhibits a bistable transition from a higher strain energy tubular state to a lower strain energy curved-crease state. Keywords: thin-walled cylinder, buckling shape control, curved-crease origami ∗
Corresponding author. Email address: [email protected] (Jiayao Ma)
Preprint submitted to International Journal of Mechanical Sciences
November 23, 2018
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Thin-walled tubes are used in many applications across a wide range of scales and disciplines. Their excellent weight-specific structural performance characteristics see them used as building elements [1, 2] and subsea fluid pipes [3, 4]. Their good crashworthiness behaviours see them used as energy absorption devices [5, 6, 7, 8]. These applications all rely on a comprehensive understanding of the buckling behaviours of thin-walled tubes, such as their stability, bifurcation and other nonlinear effects [9, 10, 11, 12]. Thin-walled cylinders are the simplest form of thin-walled tube and their behaviour under axial compression has been intensively studied for decades. It has been found that their buckling modes are strongly influenced by the material properties and geometrical parameters [13], with global and local buckling behaviours determined by length-to-diameter (L/D) and diamaterto-thickness (D/t) ratios, respectively [14, 15]. Long slender cylinders with larger L and smaller D exhibit global buckling behaviours with mode shapes arising from lateral deformation [16]. Short and medium-length cylinders have local buckling behaviours [17], where the deformation involves the formation of progressive folds within the tube itself [18]. Two significant mode shapes are seen in local buckling, based on observations of compressed thinwalled cylinders with varying D/t ratios [14]. The axi-symmetric concertina mode (also known as the ring mode) occurs with relatively thick wall thickness D/t < 90 [19, 20] and the non-symmetric diamond mode (also known as the Yoshimura mode) occurs with relatively thin wall thickness D/t > 90 [21, 22, 23]. The buckling deformation of thin-walled tubes can be controlled by utilising their imperfection sensitive characteristics [24]. This includes the use of non-uniform wall thickness for a functionally-graded form [25, 26], or employing cutouts, plastic folds, or dents on the surface to guide the deformation process to a predictable buckling mode [27, 28, 29]. Modern thin-walled tubes have also been combined with developable origami-inspired surface textures [30, 31, 32], where the plastically pre-folded creases can determine the mechanics of the buckling process and act as a mode director [33, 34]. Instead of using pre-folded geometries, an early concept has demonstrated that origamiinspired patterns can also be pre-embedded on smooth surfaces for altering the elastic buckling behaviour. Straight-crease diamond patterns, pre-embedded
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1. Introduction
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on thin-walled cylinders have been shown by the authors to generate a postbuckling configuration similar to the diamond mode [35], as shown in Figure 1. It was found that the number of circumferential and longitudinal lobes n and the slant angle α are the two most important parameters for realising inwardly-deformed ‘diamonds’, with all tested specimens with α < 45o exhibiting the diamond mode failure. However, the manifested deformation was seen to include a complex curved bending region and hence the exact deformed shape was unknown. This limitation is not unique to pre-embedded tubes; for most buckling types and modes, capturing the exact shape of a post-buckled configuration is an extremely challenging problem. Separately, curved-crease origami is a subset of origami which imparts a bending deformation in the sheet during folding [36]. Many novel engineering and architecture applications of curved-crease origami [37, 38] have all been enabled by extensive preceding work in modelling of curved-line folding [39, 40, 41]. For large elastic bending deformation, a one-dimensional analytical solution exists for a slender beams, famously known as elastica curves [42]. The elastica forms correspond to minimum bending energy configurations and also post-buckling geometries of 1D rods [43, 44, 45, 46]. The elastica curve for a simply-supported slender beam can also been utilised as the nonzero principal surface curvature for elastically-bent curved-crease origami leading to an accurate representation of a manufactured origami surface [47]. This study describes a new method to control the shape of a elastically buckled medium length thin-walled cylinder. The pre-embedded technique is used with elastica-generated curved-crease origami patterns to design a failure mode which is a modified form of the idealised diamond buckling mode. The buckling process of a thin-walled cylinder can be guided to this shape and so the exact post-buckling configuration can be accurately predetermined. Section 2 describes the generation of stabilising curved-crease origami for pre-embedding and combines higher-order elastica curve solutions with a mirror reflection method. Section 3 examines the accuracy of the predicted post-buckling configurations and the triggering conditions for a desired mode. Section 4 analyses the driving mechanics and describes energy conditions for bending under which controlled post-buckled shape can be achieved.
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Figure 1: Buckling shape control by using pre-embedded straight-crease diamond pattern. (A) 2D pattern and design parameters. (B) Undeformed cylindrical shell with design pattern. (C) Deformation when subjected to axial compression, with complex curved
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2.1. Target state curved-crease origami creation The curved-crease origami geometries utilised in this paper are first modelled at their fully-folded, or ‘target’ state. The target state of key interest in this paper is a cyclidrical tube with a pre-determined post-buckled shape. The geometric design method is summeried as follows [47]. First, a ‘higherorder’ elastica curve for a simply-supported slender rod is selected as shown in Figure 2A. It represents the exact post-buckling geometry of a compressed beam and also a minimum bending energy configuration. The curve has three length parameters: curve length L, target state support distance b, and height of the curve away from the centreline h0 . These parameters are also related to the parameter m for determining the shape of the curve, where we only consider and enumerate even modes with a central inflection point in this paper, that is m=1 is a second-mode elastica, m=2 is a fourth-mode elastica, etcetera. Specifying any three of L, b, h0 , or m gives an exact shape of a higher-order elastica curve. Extruding the curve by length w then trans-
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forms the 2D curve to a simply-supported 3D surface, as shown in Figure 2B. A higher-order mode of an elastica curve is unstable without lateral restraint, for example the m = 2 fourth-mode surface shown in Figure 2B would be expected to snap to first-mode elastica surface if it possesses only the shown end restraints. However, a pseudo-lateral restraint and stabilising effect can be provided by folding the extruded surface. The mirror reflection method is a technique for generation of a folded surface from a specified developable surface. It can be used on the extruded elastica surface by specifying a series of mirror planes and identifying folded crease lines at the intersection of the mirror planes and extruded surface. The folded surface must additionally form a tubular configuration, which introduces several conditions on the specification of mirror planes. First, to give discrete individual lobes, reflection planes are constructed within the extrusion sectional height 2h0 as shown in Figure 2C. Second, an even number of mirror planes 2n must be defined so that the final reflected surface has the same orientation as the initial surface, as shown in Figure 2D. These constraints allow plane edge angles θM B and θM A to be found with the following relationships: π(n − 1) (1) θM A = n π − θM A θM B = (2) 2 p w tan2 (θM B ) h0 = (3) 4n Sequentially truncating and reflecting the shell about the mirror planes then gives the target folded state of the tubular curved-crease origami, where n and m determine the number of circumferential and longitudinal lobes, respectively. To summarise, the target state is generated with specification of four design parameters: w, L, m, and n. Parameters n and w give h0 from Equations 1-3, from which an elastica curve can be determined with L, m, and h0 . The higher-order elastica curve forms the non-zero principal curvature for the target curved-crease surface. The tubular origami can be unrolled to generate a 2D pattern within a w × L sheet as shown in Figure 2E. The patterned sheet can either be folded into the designed target state, or simply rolled into a thin-walled cylinder with a diameter D. The latter forms a cylindrical tube, pre-embedded with
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the curved-crease pattern. The key hypothesis of this paper is that the target curved-crease origami state is the exact deformed shape of the cylindrical tube state. If so, it should be obtainable by actuating the patterned cylinder with an axial compression load and at a target displacement of lD = L − b. Subassumptions of this hypothesis include: the crease pattern is pre-embedded into the sheet without any surface pre-folding; the crease lines act as hinges and do not distort the final surface bending behaviour; and that the boundary condition is preserved as pinned-pinned as per the original elastica derivation. If this hypothesis is correct, it can be said that the post-buckled shape of a thin-walled cylinder can be precisely controlled by using a pre-embedded curved-crease origami pattern.
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Figure 2: Design procedures for tubular curved-crease origami. (A) 1D higher-order elas-
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2.2. Transformation of curved to straight-crease diamond pattern Straight-crease (SC) diamond patterns are reproduced for the comparison with curved-crease (CC) patterns in this paper. To make the two patterns comparable, the shape of target state curved lobes is transformed into di9
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amonds by connecting the lobe boundaries with straight-line segments as shown in Figure 3A-C. As the result, both patterns share a similar shape, a common sheet size, and the same number of circumferential and longitudinal lobes. Due to these characteristics, it is hypothesised that they will have a similar buckling behaviour, where the design parameter n and α strongly determine the post-bucked shape as described in [35]. These hypotheses will be systematically investigated in below sections. B.
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Figure 3: Transformation of curved and straight-crease diamond pattern. (A) Base curvedcrease 2D pattern. (B) Transformation of modular curved-lobes to diamonds. (C) Resul-
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3.1. Method To investigate the buckled shape of a patterned cylinder, nine pairs of specimens (CC/SC patterns) were manufactured and tested. These patterns were selected based on SC patterns with controllable buckling modes, tested previously by the authors [35]. They were all designed within a w × L = 278 mm × 210 mm sheet and rolled into a D/t = 294.96 cylinder with pattern design parameters summarised in Table 1. All specimens were manufactured with a t = 0.3 mm thick isotropic polypropylene sheet, which allows the requisite large elastic deformations. Crease lines were pre-embedded into the flat sheet by using a laser scoring process to reduce the material thickness along crease lines. Scored creases were approximately 0.3mm wide and 0.15-0.2mm deep on the outer tube surfaces to avoid the thickness interaction during deformation. This correspondingly reduced crease line rotational stiffnesses such that they could approximately act as ‘hinges’ during folding. Specimens were loaded under quasi-static axial compression on an Instron Universal Testing machine with a 100kN load cell. Displacement control was
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3.2. Buckling modes and force-displacement responses Tube deformations observed in tested specimens are categorised into two buckling modes based on their post-buckled shapes, summarised in Table 1. The first mode is the ‘controlled’ type as shown in Figure 4A. Patterned cylinders with this mode had all lobes buckle and bend inwards when fully compressed by lD , as per the designed curved-crease deformation mode. Patterned cylinders with such buckling mode are concluded to have a ‘shapecontrollable’ feature which is a modified form of the idealised diamond buckling mode. The second mode is the ‘uncontrolled’ type, where the patterned cylinder buckled without triggering, or only partially triggering, the pre-embedded lobes during the compression process as shown in Figure 4C. Hence, this uncontrollable mode can be considered as similar to a typical thin-walled tube local buckling behaviour. The force-displacement of selected CC/SC tube pairs is shown for controlled and uncontrolled modes in Figure 4B and D, respectively. It is seen that CC and corresponding SC tubes share a similar force-displacement response if they are undergoing the same failure mode. Five cases show CC and SC tubes with the same controllable buckling mode: m2n5, m2n6, m3n5, m3n5, and m3n8. Two cases show CC and SC tubes with the same uncontrollable buckling mode: m3n4 and m6n5. Two cases have different failures with controllable buckling mode in CC tubes and uncontrollable buckling mode in SC tubes: m1n4 and m4n9. To more closely investigate the observed differences between CC and SC buckling mode behaviours, Figure 5 shows a failure map for tested specimens, plotted against key design parameters n and α. Also included are results from [35] for SC tubes with controllable buckling mode, shown shaded in grey. SC diamond mode buckling occurs where design parameters are distributed within or close to the shaded area. The design parameter range which determines diamond mode buckling in SC patterns also determines the controllable buckling mode in CC patterns. Similarly, away from the
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used for load application, with a rate of 2 mm/min. Specimens were simplysupported between two rigid bodies, mounted on the Instron base plate and cross-head. The deformation process was captured by using a digital image correlation (DIC) system CSI Vic-3D9M, at a frame time interval of 100 milliseconds and with a 1.2/mm2 speckle pattern.
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shaded region with α close to or smaller than 45o , uncontrollable buckling mode occurs in both SC and CC tubes. Non-matching failures are shown as half-coloured dots and occur above α = 45o and close to the shaded region. This suggests that the design range for controllable buckling mode can be extended by using curved-crease origami patterns, as compared with straight-crease diamond patterns.
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Figure 4: Buckling mode classification. Left: deformation of comparable curved-crease (CC) and straight-crease (SC) patterns. Right: Force-deformation comparison of CC and
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CC and SC controlled. CC controlled, SC uncontrolled. CC and SC uncontrolled.
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Figure 5: Summary of buckling mode result comparison, where the shaded area indicates design parameters for controllable buckling mode.
Table 1: Design parameters and buckling mode comparison for patterned cylinders with
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b/L 2h0 lD Mode (-) (mm) (mm) (CC, SC) 0.988 14.39 2.45 (•, ◦) 0.982 9.03 3.88 (•, •) 0.991 6.21 1.82 (•, •) 0.888 14.39 23.56 (◦, ◦) 0.958 9.03 8.86 (•, •) 0.990 4.53 2.19 (•, •) 0.994 3.46 1.27 (•, •) 0.993 2.72 1.40 (•, ◦) 0.814 9.03 39.05 (◦, ◦)
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curved and straight-crease patterns, where • = Controlled, ◦ = Uncontrolled.
3.3. Deformed surface analysis For thin-walled cylinders achieving the controlled buckling mode, additional study was undertaken to determine the correlation between the buckled 13
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shape and the analytical folded form. The controlled buckling type m2n5 CC and SC specimens were selected for study, with their final deformed lobe geometry measured as point cloud data from DIC measurements. Only a single lobe was extracted from each specimen for the analysis, due to the large deformation influence on the light source reflection causing unmeasurable regions for the DIC system. The deformed lobe was measured at approximately 5,000 data extraction points and imported into a Rhino CAD environment. This was compared with the isolated analytical lobe highlighted in Figure 6A, with surface error calculated as the closest line distance between the data points and the analytical surface. A 6-DOF rigid body displacement optimisation routine was used to locate analytical geometry, relative to measured geometry, so as to minimise overall surface error [47]. Error measurements are plotted in Figure 6B as a contour diagram with colour legend from green to red for 0 to +2t error, and green to blue for 0 to −2t error, where sheet thickness t = 0.3mm. An average absolute surface error of 0.09 mm and 0.23 mm were seen for the CC and SC patterns, respectively. The correspondence for CC specimen demonstrates a high degree of design accuracy, where the average absolute surface error is within half of the sheet thickness. Therefore, it is concluded that the post-buckling configuration of a patterned cylinder can be accurately controlled by design of a curved-crease origami pattern with a target displacement applied. The error analysis also shows a clear difference between the SC diamond mode and CC curved-crease mode, although these are superficially very similar and were both classed above as controllable buckling modes. The substantial surface error seen for SC lobes indicates it is not collapsing to the designed geometry. Regions of high error are seen around the crease line boundary and regions of lower error are seen in the central bent lobe area. Therefore, it can be said that the curved deformation in the SC diamond mode is generated as the straight-crease pattern relaxes toward an elastica-like minimum bending energy state. The deformation process forces straight creases to somehow deform to their closest corresponding curved-crease origami. Stronger evidence can be seen in the cross section comparison shown in Figure 6C. Close correspondence to the elastica curvature is seen in the mid-region, but this reduces towards the bounding crease lines. To conclude, SC and CC pre-embedded patterns can generate a controlled buckling mode, but only elastica-generated CC patterns can give a precise geometric definition of the buckled surface. In addition, this paper has demonstrated that curved folds provide a stabilising effect for the higher-degree elastica surface shapes. This was
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4.1. Elastic Bending Energy of Deformed and Undeformed States The difference in exhibited buckling modes was hypothesised to be related to the curvature of the lobe, which changes direction and magnitude from the undeformed to the deformed state. The undeformed lobe has a uniform curvature along the circumferential direction which is obtained from the cylindrical shell surface, denoted as κ1 and shown in Figure 7A. The deformed lobe has a non-uniform curvature along the longitudinal direction which is obtained from the elastica profile, denoted as κ2 . If κ2 > κ1 , the target deformation state would have a larger surface curvature and bending strain energy potential and hence the change of bending direction may not be easily realised. However, κ1 is a uniform value and κ2 is a non-uniform function, so they cannot be compared in such a direct manner. The curvature present in both undeformed and deformed lobes can be represented as an equivalent elastic bending strain energy, assuming that either state has been folded from a flat sheet with κ1 = κ2 = 0. The equation to calculate the energy U of a sheet subjected to large elastic bending has been developed previously by the authors [35] as: Z Et3 (κ2 + κ2y + 2νκx κy )dA (4) U= 24(1 − ν 2 ) Lobe x
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hypothesised in [47], however this prior study was only able to validate a second-order elastica surface stabilised through pre-folding. The current study has validated up to an eighth-order elastica surface, stabilised through pre-embedding.
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For the polypropylene material used in this paper, Young’s Modulus E = 1,260 MPa, Poisson’s ratio ν = 0.30, and t = 0.30mm. κx and κy are the curvatures along the perpendicular directions of the lobe, so the undeformed lobe has κx = κ1 and κy = 0 and the deformed lobe has κx = 0 and κy = κ2 . The energy for undeformed and deformed lobes are denoted as U1 and U2 , respectively. Energy results for all tube configurations are summarised in Figure 7B and Table 2. All prototypes with U2 < U1 exhibited controllable buckling mode and conversely, all prototypes with U2 > U1 showed uncontrollable buckling mode. It is concluded that a pre-embedded CC pattern will generate
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• Figure 8A shows specimen m3n8 and m4n9 responses. These manifest a controlled failure mode with a smooth, approximately linear region only. • Figure 8B shows specimen m2n6 and m3n7 responses. These manifest a controlled failure with an approximately linear region followed by a slightly fluctuating plateau region. • Figure 8C shows specimen m1n4, m2n5, and m3n5 responses. These manifest a controlled failure with a classic non-linear bucking response, with a peak load followed by sharp reduction in strength and an extended plateau region. • Figure 8D shows specimen m3n4 and m6n5 responses. These manifest an uncontrolled failure but also have a classic buckling response as described for Figure 8C.
These force-displacement responses are classified Type 1, 2, and 3 behaviours for Figure 8A, B, and C-D, respectively. With reference to Figure 7B, a clear trend is seen between the relative axial displacement lD /L and the transitional behaviour. When the displacement is relatively small (lD /L = 0.006, 0.007), the lobe transition is a smooth Type 1 behaviour. With increasing displacement (lD /L = 0.009, 0.010), the buckling process is a less smooth Type 2 behaviour. When the displacement is relatively large (lD /L = 0.012, 0.018, 0.042) but with U2 < U1 , specimens undergo a controlled Type 3 behaviour. For larger displacements again, with U2 > U1 , specimens undergo an uncontrolled Type 3 behaviour. The presence of a classical buckling response indicates that a bifurcation behaviour
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4.2. Lobe Transition Characterisation Examination of the force-displacement curves of curved-crease specimens can give insight into the transitional behaviour between underformed and deformed tube shapes. Figure 8 shows the specimen responses up to their target state compression limit lD , with controlled and uncontrolled buckling modes further classified based on observed force-displacement response characteristics:
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a controlled buckling shape, if that shape has a lower elastic bending strain energy potential than the initial tubular state.
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is occurring between the tube state acting under membrane stress, and the buckled state acting under bending stress. It is hypothesised that this bifurcation is happening in all controlled specimens, but that it is more evident in specimens with a long stroke length, as a larger initial displacement and axial load can be reached in the first stable state, before transition to the second stable state.
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Figure 6: Target state surface analysis of m2n5 pattern. (A) Designed geometry with elastica surface. (B) Deformation of - top: curved-crease, bot: straight-crease pattern
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Figure 8: Force-displacement comparisons for all curved-crease specimens. They are iden-
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4.3. Lobe Transitional Energy Behaviour By assessing the complete strain energy history of a lobe, the transition and hypothesised bifurcation behaviour can be more closely studied. The m3n5 CC/SC specimens shown in Figure 4A were selected for the investigation, as they displayed the Type 3 classical non-linear buckling behaviour and so were judged likely to exhibit a clear bistability. Actual deformations were collected from the DIC system as discrete data points. These points were numerically reconstructed to a degree 5 polynomial surface, fitted with MATLAB Curve Fitting Tool. Deformations were measured and surface-fitted at 100 frame intervals from 0 %(undeformed) to 100% (deformed) state. Equation 4 was used to obtaining the surface strain energy histories, with curvatures κx and κy measured at approximately 20,000 data extraction points across the surface. Remaining parameters values in Equation 4 were as described in Section 4.1.
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Table 2: Experimental result of different types of buckling behaviours for curved-crease
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Result (Type, Mode) (3, •) (3, •) (2, •) (3, ◦) (3, •) (2, •) (1, •) (1, •) (3, ◦)
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reached the same lower energy bound. The strain energy for CC specimen remains stable afterwards, indicating the stabilised deformation was reached at a minimum of strain energy. The strain energy for SC specimens increased after the lower bound was reached, with the additional energy attributed again to straight crease lines deforming toward the curved lobe shape as shown in Figure 9. The strain energy histories support the observations of Section 3.3, with the high surface accuracy of CC specimens arising from the stable minimum strain energy state, and the relatively low surface accuracy of SC specimens arising from the straight crease line interaction with the curved lobe shape.
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In this investigation, a new post-buckled shape control technique for thin walled cylinders has been created and validated. The technique allows for control over the buckled shape of a cylindrical tube, with the shapes shown in Figure 10 all generated from the same tube by using different embedded curved-crease patterns. More significantly, the buckled shapes can be precisely described as an elastica minimum bending energy surface. A precise shape definition arose from the curved-crease geometry construction method, validated with 3D surface measurement of the deformed shape. A precise
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energy definition arose from a bending strain energy formulation based on undeformed and deformed surface curvature. The energy formulation was validated with the strain energy histories and this also showed the driving mechanics of the buckling process. Controlled buckling mode was seen to occur as a bistable transition from a higher strain energy tubular state to a lower strain energy curved-crease state. The findings of this paper also suggest numerous avenues for future research. For example, if deformed tubes are further compressed, the preembedded crease lines and generated modes are likely to have some impact on subsequent plastic deformation and energy absorption. The pre-embedded technique can potentially be applied to other types of developable tubes, such as polygonal hollow sections, tapered tubes, and tubes with different lengths. A different pre-determined configuration would be needed to reach to a permissible minimum bending energy state. These permissible states may be attainable with elastica solutions for different boundary conditions. Furthermore, the proposed shape control method might be possible with the utilisation of higher-stiffness creases. This would likely impact the peak elastic buckling load with further study needed to understand, predict, and control the peak failure load. To this end, further research on the intermediate behaviours throughout the buckling process is required, which potentially involves a robust numerical simulation method. B.
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Figure 10: Illustration of controllable buckling modes. (A) Undeformed cylindrical tube without pre-embedded patterns. (B) Pre-determined target folded shape for all controlled tubes.
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Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns Ting-Uei Lee, Xiaochen Yang, Jiayao Ma, Yan Chen, Joseph M. Gattas PII: DOI: Reference:
S0020-7403(18)33185-0 https://doi.org/10.1016/j.ijmecsci.2018.11.005 MS 4631
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
27 September 2018 5 November 2018 6 November 2018
Please cite this article as: Ting-Uei Lee, Xiaochen Yang, Jiayao Ma, Yan Chen, Joseph M. Gattas, Elastic buckling shape control of thin-walled cylinder using pre-embedded curvedcrease origami patterns, International Journal of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.11.005
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Highlights • A new elastic buckling control method for thin-walled cylinder is presented.
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• The failure mode is pre-determined as a stabilised high-order elastica surface.
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• The deformed surface is measured and shown to have a near-exact correspondence.
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Elastic buckling shape control of thin-walled cylinder using pre-embedded curved-crease origami patterns
Key Laboratory of Mechanism and Equipment Design of Ministry of Education, Tianjin, China b School of Mechanical Engineering, Tianjin University, Tianjin, China c School of Civil Engineering, University of Queensland, St. Lucia, Australia
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Ting-Uei Leea,b,c , Xiaochen Yanga,b , Jiayao Maa,b,∗, Yan Chena,b , Joseph M. Gattasb
Abstract
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Decades of research has led to a comprehensive understanding of the buckling behaviour of thin-walled tubes. Many of these studies have attempted to control the buckling-behaviour of thin-walled tubes by utilising their imperfection sensitive characteristics to guide the deformation process to a predictable buckling mode. However, a key limitation of such techniques is an inability to predict the exact deformed shape of post-buckled tubes. This study presents a new method to control the shape of a elastically buckled medium length thin-walled cylinder by using pre-embedded curved-crease origami patterns. The failure mode is pre-determined as a stabilised highorder elastica surface, which manifests via a diamond buckling mode. A set of prototypes are tested and show that the buckling process can be guided to a range of designed failure modes. The deformed surface is measured and shown to have a near-exact correspondence to the analytical description, where the average absolute surface error is less than half of the 0.3mm sheet thickness. This study then closely explores the driving mechanics of the buckling process and shows that the controllable buckling process exhibits a bistable transition from a higher strain energy tubular state to a lower strain energy curved-crease state. Keywords: thin-walled cylinder, buckling shape control, curved-crease origami ∗
Corresponding author. Email address: [email protected] (Jiayao Ma)
Preprint submitted to International Journal of Mechanical Sciences
November 23, 2018
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Thin-walled tubes are used in many applications across a wide range of scales and disciplines. Their excellent weight-specific structural performance characteristics see them used as building elements [1, 2] and subsea fluid pipes [3, 4]. Their good crashworthiness behaviours see them used as energy absorption devices [5, 6, 7, 8]. These applications all rely on a comprehensive understanding of the buckling behaviours of thin-walled tubes, such as their stability, bifurcation and other nonlinear effects [9, 10, 11, 12]. Thin-walled cylinders are the simplest form of thin-walled tube and their behaviour under axial compression has been intensively studied for decades. It has been found that their buckling modes are strongly influenced by the material properties and geometrical parameters [13], with global and local buckling behaviours determined by length-to-diameter (L/D) and diamaterto-thickness (D/t) ratios, respectively [14, 15]. Long slender cylinders with larger L and smaller D exhibit global buckling behaviours with mode shapes arising from lateral deformation [16]. Short and medium-length cylinders have local buckling behaviours [17], where the deformation involves the formation of progressive folds within the tube itself [18]. Two significant mode shapes are seen in local buckling, based on observations of compressed thinwalled cylinders with varying D/t ratios [14]. The axi-symmetric concertina mode (also known as the ring mode) occurs with relatively thick wall thickness D/t < 90 [19, 20] and the non-symmetric diamond mode (also known as the Yoshimura mode) occurs with relatively thin wall thickness D/t > 90 [21, 22, 23]. The buckling deformation of thin-walled tubes can be controlled by utilising their imperfection sensitive characteristics [24]. This includes the use of non-uniform wall thickness for a functionally-graded form [25, 26], or employing cutouts, plastic folds, or dents on the surface to guide the deformation process to a predictable buckling mode [27, 28, 29]. Modern thin-walled tubes have also been combined with developable origami-inspired surface textures [30, 31, 32], where the plastically pre-folded creases can determine the mechanics of the buckling process and act as a mode director [33, 34]. Instead of using pre-folded geometries, an early concept has demonstrated that origamiinspired patterns can also be pre-embedded on smooth surfaces for altering the elastic buckling behaviour. Straight-crease diamond patterns, pre-embedded
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on thin-walled cylinders have been shown by the authors to generate a postbuckling configuration similar to the diamond mode [35], as shown in Figure 1. It was found that the number of circumferential and longitudinal lobes n and the slant angle α are the two most important parameters for realising inwardly-deformed ‘diamonds’, with all tested specimens with α < 45o exhibiting the diamond mode failure. However, the manifested deformation was seen to include a complex curved bending region and hence the exact deformed shape was unknown. This limitation is not unique to pre-embedded tubes; for most buckling types and modes, capturing the exact shape of a post-buckled configuration is an extremely challenging problem. Separately, curved-crease origami is a subset of origami which imparts a bending deformation in the sheet during folding [36]. Many novel engineering and architecture applications of curved-crease origami [37, 38] have all been enabled by extensive preceding work in modelling of curved-line folding [39, 40, 41]. For large elastic bending deformation, a one-dimensional analytical solution exists for a slender beams, famously known as elastica curves [42]. The elastica forms correspond to minimum bending energy configurations and also post-buckling geometries of 1D rods [43, 44, 45, 46]. The elastica curve for a simply-supported slender beam can also been utilised as the nonzero principal surface curvature for elastically-bent curved-crease origami leading to an accurate representation of a manufactured origami surface [47]. This study describes a new method to control the shape of a elastically buckled medium length thin-walled cylinder. The pre-embedded technique is used with elastica-generated curved-crease origami patterns to design a failure mode which is a modified form of the idealised diamond buckling mode. The buckling process of a thin-walled cylinder can be guided to this shape and so the exact post-buckling configuration can be accurately predetermined. Section 2 describes the generation of stabilising curved-crease origami for pre-embedding and combines higher-order elastica curve solutions with a mirror reflection method. Section 3 examines the accuracy of the predicted post-buckling configurations and the triggering conditions for a desired mode. Section 4 analyses the driving mechanics and describes energy conditions for bending under which controlled post-buckled shape can be achieved.
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Figure 1: Buckling shape control by using pre-embedded straight-crease diamond pattern. (A) 2D pattern and design parameters. (B) Undeformed cylindrical shell with design pattern. (C) Deformation when subjected to axial compression, with complex curved
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2.1. Target state curved-crease origami creation The curved-crease origami geometries utilised in this paper are first modelled at their fully-folded, or ‘target’ state. The target state of key interest in this paper is a cyclidrical tube with a pre-determined post-buckled shape. The geometric design method is summeried as follows [47]. First, a ‘higherorder’ elastica curve for a simply-supported slender rod is selected as shown in Figure 2A. It represents the exact post-buckling geometry of a compressed beam and also a minimum bending energy configuration. The curve has three length parameters: curve length L, target state support distance b, and height of the curve away from the centreline h0 . These parameters are also related to the parameter m for determining the shape of the curve, where we only consider and enumerate even modes with a central inflection point in this paper, that is m=1 is a second-mode elastica, m=2 is a fourth-mode elastica, etcetera. Specifying any three of L, b, h0 , or m gives an exact shape of a higher-order elastica curve. Extruding the curve by length w then trans-
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forms the 2D curve to a simply-supported 3D surface, as shown in Figure 2B. A higher-order mode of an elastica curve is unstable without lateral restraint, for example the m = 2 fourth-mode surface shown in Figure 2B would be expected to snap to first-mode elastica surface if it possesses only the shown end restraints. However, a pseudo-lateral restraint and stabilising effect can be provided by folding the extruded surface. The mirror reflection method is a technique for generation of a folded surface from a specified developable surface. It can be used on the extruded elastica surface by specifying a series of mirror planes and identifying folded crease lines at the intersection of the mirror planes and extruded surface. The folded surface must additionally form a tubular configuration, which introduces several conditions on the specification of mirror planes. First, to give discrete individual lobes, reflection planes are constructed within the extrusion sectional height 2h0 as shown in Figure 2C. Second, an even number of mirror planes 2n must be defined so that the final reflected surface has the same orientation as the initial surface, as shown in Figure 2D. These constraints allow plane edge angles θM B and θM A to be found with the following relationships: π(n − 1) (1) θM A = n π − θM A θM B = (2) 2 p w tan2 (θM B ) h0 = (3) 4n Sequentially truncating and reflecting the shell about the mirror planes then gives the target folded state of the tubular curved-crease origami, where n and m determine the number of circumferential and longitudinal lobes, respectively. To summarise, the target state is generated with specification of four design parameters: w, L, m, and n. Parameters n and w give h0 from Equations 1-3, from which an elastica curve can be determined with L, m, and h0 . The higher-order elastica curve forms the non-zero principal curvature for the target curved-crease surface. The tubular origami can be unrolled to generate a 2D pattern within a w × L sheet as shown in Figure 2E. The patterned sheet can either be folded into the designed target state, or simply rolled into a thin-walled cylinder with a diameter D. The latter forms a cylindrical tube, pre-embedded with
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the curved-crease pattern. The key hypothesis of this paper is that the target curved-crease origami state is the exact deformed shape of the cylindrical tube state. If so, it should be obtainable by actuating the patterned cylinder with an axial compression load and at a target displacement of lD = L − b. Subassumptions of this hypothesis include: the crease pattern is pre-embedded into the sheet without any surface pre-folding; the crease lines act as hinges and do not distort the final surface bending behaviour; and that the boundary condition is preserved as pinned-pinned as per the original elastica derivation. If this hypothesis is correct, it can be said that the post-buckled shape of a thin-walled cylinder can be precisely controlled by using a pre-embedded curved-crease origami pattern.
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2.2. Transformation of curved to straight-crease diamond pattern Straight-crease (SC) diamond patterns are reproduced for the comparison with curved-crease (CC) patterns in this paper. To make the two patterns comparable, the shape of target state curved lobes is transformed into di9
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amonds by connecting the lobe boundaries with straight-line segments as shown in Figure 3A-C. As the result, both patterns share a similar shape, a common sheet size, and the same number of circumferential and longitudinal lobes. Due to these characteristics, it is hypothesised that they will have a similar buckling behaviour, where the design parameter n and α strongly determine the post-bucked shape as described in [35]. These hypotheses will be systematically investigated in below sections. B.
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Figure 3: Transformation of curved and straight-crease diamond pattern. (A) Base curvedcrease 2D pattern. (B) Transformation of modular curved-lobes to diamonds. (C) Resul-
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3.1. Method To investigate the buckled shape of a patterned cylinder, nine pairs of specimens (CC/SC patterns) were manufactured and tested. These patterns were selected based on SC patterns with controllable buckling modes, tested previously by the authors [35]. They were all designed within a w × L = 278 mm × 210 mm sheet and rolled into a D/t = 294.96 cylinder with pattern design parameters summarised in Table 1. All specimens were manufactured with a t = 0.3 mm thick isotropic polypropylene sheet, which allows the requisite large elastic deformations. Crease lines were pre-embedded into the flat sheet by using a laser scoring process to reduce the material thickness along crease lines. Scored creases were approximately 0.3mm wide and 0.15-0.2mm deep on the outer tube surfaces to avoid the thickness interaction during deformation. This correspondingly reduced crease line rotational stiffnesses such that they could approximately act as ‘hinges’ during folding. Specimens were loaded under quasi-static axial compression on an Instron Universal Testing machine with a 100kN load cell. Displacement control was
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3.2. Buckling modes and force-displacement responses Tube deformations observed in tested specimens are categorised into two buckling modes based on their post-buckled shapes, summarised in Table 1. The first mode is the ‘controlled’ type as shown in Figure 4A. Patterned cylinders with this mode had all lobes buckle and bend inwards when fully compressed by lD , as per the designed curved-crease deformation mode. Patterned cylinders with such buckling mode are concluded to have a ‘shapecontrollable’ feature which is a modified form of the idealised diamond buckling mode. The second mode is the ‘uncontrolled’ type, where the patterned cylinder buckled without triggering, or only partially triggering, the pre-embedded lobes during the compression process as shown in Figure 4C. Hence, this uncontrollable mode can be considered as similar to a typical thin-walled tube local buckling behaviour. The force-displacement of selected CC/SC tube pairs is shown for controlled and uncontrolled modes in Figure 4B and D, respectively. It is seen that CC and corresponding SC tubes share a similar force-displacement response if they are undergoing the same failure mode. Five cases show CC and SC tubes with the same controllable buckling mode: m2n5, m2n6, m3n5, m3n5, and m3n8. Two cases show CC and SC tubes with the same uncontrollable buckling mode: m3n4 and m6n5. Two cases have different failures with controllable buckling mode in CC tubes and uncontrollable buckling mode in SC tubes: m1n4 and m4n9. To more closely investigate the observed differences between CC and SC buckling mode behaviours, Figure 5 shows a failure map for tested specimens, plotted against key design parameters n and α. Also included are results from [35] for SC tubes with controllable buckling mode, shown shaded in grey. SC diamond mode buckling occurs where design parameters are distributed within or close to the shaded area. The design parameter range which determines diamond mode buckling in SC patterns also determines the controllable buckling mode in CC patterns. Similarly, away from the
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used for load application, with a rate of 2 mm/min. Specimens were simplysupported between two rigid bodies, mounted on the Instron base plate and cross-head. The deformation process was captured by using a digital image correlation (DIC) system CSI Vic-3D9M, at a frame time interval of 100 milliseconds and with a 1.2/mm2 speckle pattern.
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shaded region with α close to or smaller than 45o , uncontrollable buckling mode occurs in both SC and CC tubes. Non-matching failures are shown as half-coloured dots and occur above α = 45o and close to the shaded region. This suggests that the design range for controllable buckling mode can be extended by using curved-crease origami patterns, as compared with straight-crease diamond patterns.
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Figure 4: Buckling mode classification. Left: deformation of comparable curved-crease (CC) and straight-crease (SC) patterns. Right: Force-deformation comparison of CC and
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Table 1: Design parameters and buckling mode comparison for patterned cylinders with
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b/L 2h0 lD Mode (-) (mm) (mm) (CC, SC) 0.988 14.39 2.45 (•, ◦) 0.982 9.03 3.88 (•, •) 0.991 6.21 1.82 (•, •) 0.888 14.39 23.56 (◦, ◦) 0.958 9.03 8.86 (•, •) 0.990 4.53 2.19 (•, •) 0.994 3.46 1.27 (•, •) 0.993 2.72 1.40 (•, ◦) 0.814 9.03 39.05 (◦, ◦)
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3.3. Deformed surface analysis For thin-walled cylinders achieving the controlled buckling mode, additional study was undertaken to determine the correlation between the buckled 13
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shape and the analytical folded form. The controlled buckling type m2n5 CC and SC specimens were selected for study, with their final deformed lobe geometry measured as point cloud data from DIC measurements. Only a single lobe was extracted from each specimen for the analysis, due to the large deformation influence on the light source reflection causing unmeasurable regions for the DIC system. The deformed lobe was measured at approximately 5,000 data extraction points and imported into a Rhino CAD environment. This was compared with the isolated analytical lobe highlighted in Figure 6A, with surface error calculated as the closest line distance between the data points and the analytical surface. A 6-DOF rigid body displacement optimisation routine was used to locate analytical geometry, relative to measured geometry, so as to minimise overall surface error [47]. Error measurements are plotted in Figure 6B as a contour diagram with colour legend from green to red for 0 to +2t error, and green to blue for 0 to −2t error, where sheet thickness t = 0.3mm. An average absolute surface error of 0.09 mm and 0.23 mm were seen for the CC and SC patterns, respectively. The correspondence for CC specimen demonstrates a high degree of design accuracy, where the average absolute surface error is within half of the sheet thickness. Therefore, it is concluded that the post-buckling configuration of a patterned cylinder can be accurately controlled by design of a curved-crease origami pattern with a target displacement applied. The error analysis also shows a clear difference between the SC diamond mode and CC curved-crease mode, although these are superficially very similar and were both classed above as controllable buckling modes. The substantial surface error seen for SC lobes indicates it is not collapsing to the designed geometry. Regions of high error are seen around the crease line boundary and regions of lower error are seen in the central bent lobe area. Therefore, it can be said that the curved deformation in the SC diamond mode is generated as the straight-crease pattern relaxes toward an elastica-like minimum bending energy state. The deformation process forces straight creases to somehow deform to their closest corresponding curved-crease origami. Stronger evidence can be seen in the cross section comparison shown in Figure 6C. Close correspondence to the elastica curvature is seen in the mid-region, but this reduces towards the bounding crease lines. To conclude, SC and CC pre-embedded patterns can generate a controlled buckling mode, but only elastica-generated CC patterns can give a precise geometric definition of the buckled surface. In addition, this paper has demonstrated that curved folds provide a stabilising effect for the higher-degree elastica surface shapes. This was
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4.1. Elastic Bending Energy of Deformed and Undeformed States The difference in exhibited buckling modes was hypothesised to be related to the curvature of the lobe, which changes direction and magnitude from the undeformed to the deformed state. The undeformed lobe has a uniform curvature along the circumferential direction which is obtained from the cylindrical shell surface, denoted as κ1 and shown in Figure 7A. The deformed lobe has a non-uniform curvature along the longitudinal direction which is obtained from the elastica profile, denoted as κ2 . If κ2 > κ1 , the target deformation state would have a larger surface curvature and bending strain energy potential and hence the change of bending direction may not be easily realised. However, κ1 is a uniform value and κ2 is a non-uniform function, so they cannot be compared in such a direct manner. The curvature present in both undeformed and deformed lobes can be represented as an equivalent elastic bending strain energy, assuming that either state has been folded from a flat sheet with κ1 = κ2 = 0. The equation to calculate the energy U of a sheet subjected to large elastic bending has been developed previously by the authors [35] as: Z Et3 (κ2 + κ2y + 2νκx κy )dA (4) U= 24(1 − ν 2 ) Lobe x
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hypothesised in [47], however this prior study was only able to validate a second-order elastica surface stabilised through pre-folding. The current study has validated up to an eighth-order elastica surface, stabilised through pre-embedding.
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For the polypropylene material used in this paper, Young’s Modulus E = 1,260 MPa, Poisson’s ratio ν = 0.30, and t = 0.30mm. κx and κy are the curvatures along the perpendicular directions of the lobe, so the undeformed lobe has κx = κ1 and κy = 0 and the deformed lobe has κx = 0 and κy = κ2 . The energy for undeformed and deformed lobes are denoted as U1 and U2 , respectively. Energy results for all tube configurations are summarised in Figure 7B and Table 2. All prototypes with U2 < U1 exhibited controllable buckling mode and conversely, all prototypes with U2 > U1 showed uncontrollable buckling mode. It is concluded that a pre-embedded CC pattern will generate
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• Figure 8A shows specimen m3n8 and m4n9 responses. These manifest a controlled failure mode with a smooth, approximately linear region only. • Figure 8B shows specimen m2n6 and m3n7 responses. These manifest a controlled failure with an approximately linear region followed by a slightly fluctuating plateau region. • Figure 8C shows specimen m1n4, m2n5, and m3n5 responses. These manifest a controlled failure with a classic non-linear bucking response, with a peak load followed by sharp reduction in strength and an extended plateau region. • Figure 8D shows specimen m3n4 and m6n5 responses. These manifest an uncontrolled failure but also have a classic buckling response as described for Figure 8C.
These force-displacement responses are classified Type 1, 2, and 3 behaviours for Figure 8A, B, and C-D, respectively. With reference to Figure 7B, a clear trend is seen between the relative axial displacement lD /L and the transitional behaviour. When the displacement is relatively small (lD /L = 0.006, 0.007), the lobe transition is a smooth Type 1 behaviour. With increasing displacement (lD /L = 0.009, 0.010), the buckling process is a less smooth Type 2 behaviour. When the displacement is relatively large (lD /L = 0.012, 0.018, 0.042) but with U2 < U1 , specimens undergo a controlled Type 3 behaviour. For larger displacements again, with U2 > U1 , specimens undergo an uncontrolled Type 3 behaviour. The presence of a classical buckling response indicates that a bifurcation behaviour
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4.2. Lobe Transition Characterisation Examination of the force-displacement curves of curved-crease specimens can give insight into the transitional behaviour between underformed and deformed tube shapes. Figure 8 shows the specimen responses up to their target state compression limit lD , with controlled and uncontrolled buckling modes further classified based on observed force-displacement response characteristics:
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a controlled buckling shape, if that shape has a lower elastic bending strain energy potential than the initial tubular state.
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is occurring between the tube state acting under membrane stress, and the buckled state acting under bending stress. It is hypothesised that this bifurcation is happening in all controlled specimens, but that it is more evident in specimens with a long stroke length, as a larger initial displacement and axial load can be reached in the first stable state, before transition to the second stable state.
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Figure 6: Target state surface analysis of m2n5 pattern. (A) Designed geometry with elastica surface. (B) Deformation of - top: curved-crease, bot: straight-crease pattern
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4.3. Lobe Transitional Energy Behaviour By assessing the complete strain energy history of a lobe, the transition and hypothesised bifurcation behaviour can be more closely studied. The m3n5 CC/SC specimens shown in Figure 4A were selected for the investigation, as they displayed the Type 3 classical non-linear buckling behaviour and so were judged likely to exhibit a clear bistability. Actual deformations were collected from the DIC system as discrete data points. These points were numerically reconstructed to a degree 5 polynomial surface, fitted with MATLAB Curve Fitting Tool. Deformations were measured and surface-fitted at 100 frame intervals from 0 %(undeformed) to 100% (deformed) state. Equation 4 was used to obtaining the surface strain energy histories, with curvatures κx and κy measured at approximately 20,000 data extraction points across the surface. Remaining parameters values in Equation 4 were as described in Section 4.1.
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The analysis results are shown in Figure 9, plotted as strain energy frame interval measurements versus relative deformation. It can be seen that analytical predictions for U1 and U2 CC specimen strain energies have a good correspondence with the measured experimental (EXP) strain energy. This indicates the analysis and analytical methods are reliable. Both SC and CC specimens demonstrate a classic elastic snap-through behaviour at approximately 25-30% relative deformation. This is characterised by an initial stable state, here the tube energy state U1 , from which the strain energy increases as the tube is loaded. There is a sudden energy drop as the deformation snaps to a second stable state with a lower energy potential, here the curved-crease energy state U2 . Prior to the snap-through, it can be noticed that the SC specimen has a higher energy barrier than the CC specimen, which is attributed to additional strain energy needed to deform the straight creases to allow a curved-lobe snap through. The bistability behaviour of the CC patterns is therefore relatively easier to trigger, which gives a reasonable explanation of the wider design range of curved-crease origami patterns which exhibited controlled buckling mode as highlighted in Section 3.2. Following the snap-through, it can also be noticed that the strain energy histories of the CC and SC specimens are very different. Both specimens have
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reached the same lower energy bound. The strain energy for CC specimen remains stable afterwards, indicating the stabilised deformation was reached at a minimum of strain energy. The strain energy for SC specimens increased after the lower bound was reached, with the additional energy attributed again to straight crease lines deforming toward the curved lobe shape as shown in Figure 9. The strain energy histories support the observations of Section 3.3, with the high surface accuracy of CC specimens arising from the stable minimum strain energy state, and the relatively low surface accuracy of SC specimens arising from the straight crease line interaction with the curved lobe shape.
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In this investigation, a new post-buckled shape control technique for thin walled cylinders has been created and validated. The technique allows for control over the buckled shape of a cylindrical tube, with the shapes shown in Figure 10 all generated from the same tube by using different embedded curved-crease patterns. More significantly, the buckled shapes can be precisely described as an elastica minimum bending energy surface. A precise shape definition arose from the curved-crease geometry construction method, validated with 3D surface measurement of the deformed shape. A precise
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energy definition arose from a bending strain energy formulation based on undeformed and deformed surface curvature. The energy formulation was validated with the strain energy histories and this also showed the driving mechanics of the buckling process. Controlled buckling mode was seen to occur as a bistable transition from a higher strain energy tubular state to a lower strain energy curved-crease state. The findings of this paper also suggest numerous avenues for future research. For example, if deformed tubes are further compressed, the preembedded crease lines and generated modes are likely to have some impact on subsequent plastic deformation and energy absorption. The pre-embedded technique can potentially be applied to other types of developable tubes, such as polygonal hollow sections, tapered tubes, and tubes with different lengths. A different pre-determined configuration would be needed to reach to a permissible minimum bending energy state. These permissible states may be attainable with elastica solutions for different boundary conditions. Furthermore, the proposed shape control method might be possible with the utilisation of higher-stiffness creases. This would likely impact the peak elastic buckling load with further study needed to understand, predict, and control the peak failure load. To this end, further research on the intermediate behaviours throughout the buckling process is required, which potentially involves a robust numerical simulation method. B.
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510
[42] L. Euler, Methodus inveniendi lineas curvas maxima minimive proprietate gaudentes additamentum i, De Curvis Elasticis, Lausanne and Geneva 1744.
508
26